Building on the analogy between Feynman graph polynomials and those of electrical circuits, we then formulated a second class of parametric representations. For these, the integration variables represent the effective resistances between vertices of the simplex,
rather than
the conductivities
(i.e., the inverse Schwinger parameters) used previously.
This change of variables immediately diagonalises the Schwinger exponential, expressing the \(n\) -point function as a standard Laplace transform of a product of polynomials raised to generalised powers. These polynomials correspond to the determinant and first minors of the Cayley-Menger matrix for the simplex, which
plays an analogous role to the Gram matrix for this second class of parametrisations. From the form of these polynomials, new weight-shifting operators can immediately be constructed to raise the power of these polynomials, with further shift operators following by shadow conjugation. Besides shifting the scaling dimensions of external operators, these new weight-shifting operators raise the spacetime dimension by two.
They therefore generalise the 3-point shift operators of [1]}, [2]} to \(n\) -points, and
constitute a distinct class of operators to those
identified in [3]}.
